Let me be the doubting Thomas here...
-The tree is in the backyard, no way to get a vehicle back there (MAYBE one of those mini bobcats could fit through the side yard).
-Half the roots are on other side of fence/someone else's property.
-Established tree will have spreading root system, no way you can save it all, will be losing a lot of the roots which negates that advantage of getting a mature tree.
-To fit through the side yard you may have to cut it back heavily, negating advantage of getting a large tree
-Then the tree will be in possible shock and growth may be stunted for a few years, negating advantage of mature tree
I posted an ad similar to this guys when I moved in my house. I got rid of all the ornamental landscaping, put "free bushes" on CL and people came, spending hours of hard labor digging out these small bushes with well established root systems. Saved me a lot of work!
Good luck though, I guess if you are determined you will find a way!